Let $F \subseteq k$ be fields, $A$ an affine $k$-algebra, and $A_0$ an $F$-structure on $A$ (that is, an $F$-subalgebra of $A$ which is finitely generated as an $F$-algebra for which the natural $k$-algebra homomorphism $k \otimes_F A_0 \rightarrow A$ is an isomorphism). We already have the Zariski topology on the set $Max(A)$ of maximal ideals of $A$.
With respect to this $F$-structure $A_0$, we can define a coarser topology on $Max(A)$, wherein we say a closed set $X \subseteq Max(A)$ is $F$-closed if the radical ideal $I(X)$ is generated (as an ideal, $k$-module, doesn't matter) by elements of $A_0$. In other words, with respect to the ring monomorphism $\phi: A_0 \rightarrow A$, the radical ideal $I(X)$ is extended.
I'm trying to show that the $F$-closed sets define a topology on $Max(A)$, but it's not going well. If $V_j$ is a collection of $F$-closed sets of $Max(A)$, then the intersection $\bigcap\limits_j V_j$ corresponds to the sum of extended ideals $\sum\limits_j I(V_j)$. A sum of extended ideals is extended, no problem. On the other hand, $I$ applied to a union of two $F$-closed sets $V_1, V_2$ corresponds to the intersection $I(V_1) \cap I(V_2)$. But in general, an intersection of extended ideals need not be extended. What should I do?
$A$ is $A_0$-flat (since $k$ is $F$-flat), and then the intersection commutes with the extension of ideals (see Matsumura, CRT, Theorem 7.4(ii)).